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Is there a forcing notion $\mathbb{P}$ such that:

1. For any $p \in\mathbb{P}, \mathbb{P}/p = \{q \in \mathbb{P}: q \leq p \}$ is not forcing isomorphic to a homogeneous forcing notion,

2. For all $G$, $\mathbb{P}$-generic over $V$, $HOD^{V[G]} \subseteq V$.

Here by homogeneity I mean cone homogeneity, i.e. for any p, q in $\mathbb{P}$, there are $p' \leq p, q' \leq q$ and an isomorphism from $\mathbb{P}/ p' $ onto $\mathbb{P}/q'.$

Is there a separative forcing notion $\mathbb{P}$ such that:

1. For any $p \in\mathbb{P}, \mathbb{P}/p = \{q \in \mathbb{P}: q \leq p \}$ is not forcing isomorphic to any homogeneous forcing notion,

2. For all $G$, $\mathbb{P}$-generic over $V$, $HOD^{V[G]} \subseteq V$.

Here by homogeneity I mean cone homogeneity, i.e. for any p, q in $\mathbb{P}$, there are $p' \leq p, q' \leq q$ and an isomorphism from $\mathbb{P}/ p' $ onto $\mathbb{P}/q'.$

Is there a forcing notion $\mathbb{P}$ such that:

1. For any $p \in\mathbb{P}, \mathbb{P}/p = \{q \in \mathbb{P}: q \leq p \}$ is not forcing isomorphic to a homogeneous forcing notion,

2. For all $G$, $\mathbb{P}$-generic over $V$, $HOD^{V[G]} \subseteq V$.

Is there a forcing notion $\mathbb{P}$ such that:

1. For any $p \in\mathbb{P}, \mathbb{P}/p = \{q \in \mathbb{P}: q \leq p \}$ is not forcing isomorphic to a homogeneous forcing notion,

2. For all $G$, $\mathbb{P}$-generic over $V$, $HOD^{V[G]} \subseteq V$.

Here by homogeneity I mean cone homogeneity, i.e. for any p, q in $\mathbb{P}$, there are $p' \leq p, q' \leq q$ and an isomorphism from $\mathbb{P}/ p' $ onto $\mathbb{P}/q'.$

Is there a forcing notion $\mathbb{P}$ such that:

1. For any $p \in\mathbb{P}, \mathbb{P}/p = \{q \in \mathbb{P}: q \leq p \}$ is not forcing isomorphic to a homogeneous forcing notion,

2. For all $G$, $\mathbb{P}$-generic over $V$, $HOD^{V[G]} \subseteq V$.

If so, then can we also require the following extra assumption.

3. For all $G, H$, $\mathbb{P}$-generic over $V, V[G] \equiv V[H]$.

Is there a forcing notion $\mathbb{P}$ such that:

1. For any $p \in\mathbb{P}, \mathbb{P}/p = \{q \in \mathbb{P}: q \leq p \}$ is not forcing isomorphic to a homogeneous forcing notion,

2. For all $G$, $\mathbb{P}$-generic over $V$, $HOD^{V[G]} \subseteq V$.